In robustness analysis, hypotheses are supported to the extent that a result proves robust, and a result is robust to the extent that we detect it in diverse ways. But what precise sense of diversity is at work here? In this paper, I show that the formal explications of evidential diversity most often appealed to in work on robustness – which all draw in one way or another on probabilistic independence – fail to shed light on the notion (...) of diversity relevant to robustness analysis. I close by briefly outlining a promising alternative approach inspired by Horwich’s (1982) eliminative account of evidential diversity. (shrink)

Suppose several individuals (e.g., experts on a panel) each assign probabilities to some events. How can these individual probability assignments be aggregated into a single collective probability assignment? This article reviews several proposed solutions to this problem. We focus on three salient proposals: linear pooling (the weighted or unweighted linear averaging of probabilities), geometric pooling (the weighted or unweighted geometric averaging of probabilities), and multiplicative pooling (where probabilities are multiplied rather than averaged). We present axiomatic characterisations of each class of (...) pooling functions (most of them classic, but one new) and argue that linear pooling can be justified procedurally, but not epistemically, while the other two pooling methods can be justified epistemically. The choice between them, in turn, depends on whether the individuals' probability assignments are based on shared information or on private information. We conclude by mentioning a number of other pooling methods. (shrink)

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We (...) formalize this idea by introducing a 'premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

The epistemic modal auxiliaries must and might are vehicles for expressing the force with which a proposition follows from some body of evidence or information. Standard approaches model these operators using quantificational modal logic, but probabilistic approaches are becoming increasingly influential. According to a traditional view, must is a maximally strong epistemic operator and might is a bare possibility one. A competing account—popular amongst proponents of a probabilisitic turn—says that, given a body of evidence, must \ entails that \\) (...) is high but non-maximal and might \ that \\) is significantly greater than 0. Drawing on several observations concerning the behavior of must, might and similar epistemic operators in evidential contexts, deductive inferences, downplaying and retractions scenarios, and expressions of epistemic tension, I argue that those two influential accounts have systematic descriptive shortcomings. To better make sense of their complex behavior, I propose instead a broadly Kratzerian account according to which must \ entails that \ = 1\) and might \ that \ > 0\), given a body of evidence and a set of normality assumptions about the world. From this perspective, must and might are vehicles for expressing a common mode of reasoning whereby we draw inferences from specific bits of evidence against a rich set of background assumptions—some of which we represent as defeasible—which capture our general expectations about the world. I will show that the predictions of this Kratzerian account can be substantially refined once it is combined with a specific yet independently motivated ‘grammatical’ approach to the computation of scalar implicatures. Finally, I discuss some implications of these results for more general discussions concerning the empirical and theoretical motivation to adopt a probabilisitic semantic framework. (shrink)

Two recent and influential papers, van Rooij 2007 and Lassiter 2012, propose solutions to the proviso problem that make central use of related notions of independence—qualitative in the first case, probabilistic in the second. We argue here that, if these solutions are to work, they must incorporate an implicit assumption about presupposition accommodation, namely that accommodation does not interfere with existing qualitative or probabilistic independencies. We show, however, that this assumption is implausible, as updating beliefs with conditional (...) information does not in general preserve independencies. We conclude that the approach taken by van Rooij and Lassiter does not succeed in resolving the proviso problem. (shrink)

Robustness arguments hold that hypotheses are more likely to be true when they are confirmed by diverse kinds of evidence. Robustness arguments require the confirming evidence to be independent. We identify two kinds of independence appealed to in robustness arguments: ontic independence —when the multiple lines of evidence depend on different materials, assumptions, or theories—and probabilistic independence. Many assume that OI is sufficient for a robustness argument to be warranted. However, we argue that, as typically construed, (...) OI is not a sufficient independence condition for warranting robustness arguments. We show that OI evidence can collectively confirm a hypothesis to a lower degree than individual lines of evidence, contrary to the standard assumption undergirding usual robustness arguments. We employ Bayesian networks to represent the ideal empirical scenario for a robustness argument and a variety of ways in which empirical scenarios can fall short of this ideal. (shrink)

Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic (...) knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch bookability iff they have consistent probabilistic beliefs. (shrink)

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it introduces (...) a novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

On explanationist accounts of genealogical debunking, roughly, a belief is debunked when its explanation is not suitably related to its content. We argue that explanationism cannot accommodate cases in which beliefs are explained by factors unrelated to their contents but are nonetheless independently justified. Justification-specific versions of explanationism face an iteration of the problem. The best account of debunking is a probabilistic account according to which subject S’s justification J for their belief that P is debunked when S learns (...) that J is no more likely to be true on the hypothesis that P than on the hypothesis that ¬P. The probabilistic criterion is fully general, applying not only to cases where the learned undercutting defeater is a proposition about our beliefs or other mental states but to any case of undercutting defeat, providing the grounds for a debunking argument against the existence of a special, metacognitive debunking principle. (shrink)

We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are (...) aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other. (shrink)

Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected (...) by environmental factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed. (shrink)

We consider two ways one might use algorithmic randomness to characterize a probabilistic law. The first is a generative chance* law. Such laws involve a nonstandard notion of chance. The second is a probabilistic* constraining law. Such laws impose relative frequency and randomness constraints that every physically possible world must satisfy. While each notion has virtues, we argue that the latter has advantages over the former. It supports a unified governing account of non-Humean laws and provides independently motivated (...) solutions to issues in the Humean best-system account. On both notions, we have a much tighter connection between probabilistic laws and their corresponding sets of possible worlds. Certain histories permitted by traditional probabilistic laws are ruled out as physically impossible. As a result, such laws avoid one variety of empirical underdetermination, but the approach reveals other varieties of underdetermination that are typically overlooked. (shrink)

Conditional independence tests have received special attention lately in machine learning and computational intelligence related literature as an important indicator of the relationship among the variables used by their models. In the field of probabilistic graphical models, which includes Bayesian network models, conditional independence tests are especially important for the task of learning the probabilistic graphical model structure from data. In this paper, we propose the full Bayesian significance test for tests of conditional independence for (...) discrete datasets. The full Bayesian significance test is a powerful Bayesian test for precise hypothesis, as an alternative to the frequentist’s significance tests (characterized by the calculation of the p-value). (shrink)

Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically (...) independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. (shrink)

One thousand fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads. I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a (...) central role in their theorizing. (shrink)

Causation is a macroscopic phenomenon. The temporal asymmetry displayed by causation must somehow emerge along with other asymmetric macroscopic phenomena like entropy increase and the arrow of radiation. I shall approach this issue by considering ‘causal inference’ techniques that allow causal relations to be inferred from sets of observed correlations. I shall show that these techniques are best explained by a reduction of causation to structures of equations with probabilistically independent exogenous terms. This exogenous probabilistic independence imposes a (...) recursive order on these equations and a consequent distinction between dependent and independent variables that lines up with the temporal asymmetry of causation. (shrink)

The aim of this dissertation is to comprehensively study various robustness arguments proposed in the literature from Levins to Lloyd as well as the opposition offered to them and pose enquiry into the degree of epistemic virtue that they provide to the model prediction results with respect to climate science and modeling. Another critical issue that this dissertation strives to examine is that of the actual epistemic notion that is operational when scientists and philosophers appeal to robustness. In attempting to (...) explicate this idea, the discussion turns to arguments provided by Schupbach who completely rejects probabilistic independence in favour of explanatory reasoning, Stegenga and Menon who still see some value in probabilistic independence, and Winsberg who takes applies Schupbach’s to climate science, going beyond models to involve multi-modal evidence. After an exhaustive discussion on these arguments, this dissertation attempts to provide a thorough and updated notion of robustness in climate modeling and climate science. (shrink)

to appear in Lambert, E. and J. Schwenkler (eds.) Transformative Experience (OUP) -/- L. A. Paul (2014, 2015) argues that the possibility of epistemically transformative experiences poses serious and novel problems for the orthodox theory of rational choice, namely, expected utility theory — I call her argument the Utility Ignorance Objection. In a pair of earlier papers, I responded to Paul’s challenge (Pettigrew 2015, 2016), and a number of other philosophers have responded in similar ways (Dougherty, et al. 2015, Harman (...) 2015) — I call our argument the Fine-Graining Response. Paul has her own reply to this response, which we might call the Authenticity Reply. But Sarah Moss has recently offered an alternative reply to the Fine-Graining Response on Paul’s behalf (Moss 2017) — we’ll call it the No Knowledge Reply. This appeals to the knowledge norm of action, together with Moss’ novel and intriguing account of probabilistic knowledge. In this paper, I consider Moss’ reply and argue that it fails. I argue first that it fails as a reply made on Paul’s behalf, since it forces us to abandon many of the features of Paul’s challenge that make it distinctive and with which Paul herself is particularly concerned. Then I argue that it fails as a reply independent of its fidelity to Paul’s intentions. (shrink)

Decision-making typically requires judgments about causal relations: we need to know the causal effects of our actions and the causal relevance of various environmental factors. We investigate how several individuals' causal judgments can be aggregated into collective causal judgments. First, we consider the aggregation of causal judgments via the aggregation of probabilistic judgments, and identify the limitations of this approach. We then explore the possibility of aggregating causal judgments independently of probabilistic ones. Formally, we introduce the problem of (...) causal-network aggregation. Finally, we revisit the aggregation of probabilistic judgments when this is constrained by prior aggregation of qualitative causal judgments. (shrink)

The problem of indeterminism in quantum mechanics usually being considered as a generalization determinism of classical mechanics and physics for the case of discrete (quantum) changes is interpreted as an only mathematical problem referring to the relation of a set of independent choices to a well-ordered series therefore regulated by the equivalence of the axiom of choice and the well-ordering “theorem”. The former corresponds to quantum indeterminism, and the latter, to classical determinism. No other premises (besides the above only mathematical (...) equivalence) are necessary to explain how the probabilistic causation of quantum mechanics refers to the unambiguous determinism of classical physics. The same equivalence underlies the mathematical formalism of quantum mechanics. It merged the well-ordered components of the vectors of Heisenberg’s matrix mechanics and the non-ordered members of the wave functions of Schrödinger’s undulatory mechanics. The mathematical condition of that merging is just the equivalence of the axiom of choice and the well-ordering theorem implying in turn Max Born’s probabilistic interpretation of quantum mechanics. Particularly, energy conservation is justified differently than classical physics. It is due to the equivalence at issue rather than to the principle of least action. One may involve two forms of energy conservation corresponding whether to the smooth changes of classical physics or to the discrete changes of quantum mechanics. Further both kinds of changes can be equated to each other under the unified energy conservation as well as the conditions for the violation of energy conservation to be investigated therefore directing to a certain generalization of energy conservation. (shrink)

On analogy with testimony, I define a notion of a scientific theory’s lacking or having candor, in a testing situation, according to whether the theory under test is probabilistically relevant to the processes in the test procedures, and thereby to the reliability of test outcomes. I argue that this property identifies what is distinctive about those theories that Karl Popper denounced as exhibiting “reinforced dogmatism” through their self-protective behavior (e.g., psychoanalysis, Hegelianism, Marxism). I explore whether lack of candor interferes with (...) the testing of theories, and conclude that (1) our default attitude toward theories that lack candor in a given test should be suspicion, but (2) the circumstance that a theory lacks candor in a testing situation does not preclude obtaining independent evidence for the auxiliary assumptions to which the theory is probabilistically relevant, and thereby eliminating the problem that lack of candor creates. Thus, Popper was right to think that lack of candor is a bad thing, but wrong to conclude that candor is a criterion of the scientificity of a theory. Seeing this requires recognition of some differences between intuitive relevance and probabilistic relevance, and proper appreciation of the notion of screening off and of the fact that probabilistic relevance is not transitive. (shrink)

The coherence of independent reports provides a strong reason to believe that the reports are true. This plausible claim has come under attack from recent work in Bayesian epistemology. This work shows that, under certain probabilistic conditions, coherence cannot increase the probability of the target claim. These theorems are taken to demonstrate that epistemic coherentism is untenable. To date no one has investigated how these results bear on different conceptions of coherence. I investigate this situation using Thagard’s ECHO model (...) of explanatory coherence. Thagard’s ECHO model provides a natural representation of the evidential significance of multiple independent reports. (shrink)

Evidentialists say that a necessary condition of sound epistemic reasoning is that our beliefs reflect only our evidence. This thesis arguably conflicts with standard Bayesianism, due to the importance of prior probabilities in the latter. Some evidentialists have responded by modelling belief-states using imprecise probabilities (Joyce 2005). However, Roger White (2010) and Aron Vallinder (2018) argue that this Imprecise Bayesianism is incompatible with evidentialism due to “inertia”, where Imprecise Bayesian agents become stuck in a state of ambivalence towards hypotheses. Additionally, (...) escapes from inertia apparently only create further conflicts with evidentialism. This dilemma gives a reason for evidentialist imprecise probabilists to look for alternatives without inertia. I shall argue that Henry E. Kyburg’s approach offers an evidentialist-friendly imprecise probability theory without inertia, and that its relevant anti-inertia features are independently justified. I also connect the traditional epistemological debates concerning the “ethics of belief” more systematically with formal epistemology than has been hitherto done. (shrink)

Laws of nature seem to take two forms. Fundamental physics discovers laws that hold without exception, ‘strict laws’, as they are sometimes called; even if some laws of fundamental physics are irreducibly probabilistic, the probabilistic relation is thought not to waver. In the nonfundamental, or special, sciences, matters differ. Laws of such sciences as psychology and economics hold only ceteris paribus – that is, when other things are equal. Sometimes events accord with these ceteris paribus laws (c.p. laws, (...) hereafter), but sometimes the laws are not manifest, as if they have somehow been placed in abeyance: the regular relation indicative of natural law can fail in circumstances where an analogous outcome would effectively refute the assertion of strict law. Many authors have questioned the supposed distinction between strict laws and c.p. laws. The brief against it comprises various considerations: from the complaint that c.p. clauses are void of meaning to the claim that, although understood well enough, they should appear in all law-statements. These two concerns, among others, are addressed in due course, but first, I venture a positive proposal. I contend that there is an important contrast between strict laws and c.p. laws, one that rests on an independent distinction between combinatorial and noncombinatorial nomic principles.2 Instantiations of certain properties, e.g., mass and charge, nomically produce individual forces, or more generally, causal influences,3 in accordance with noncombinatorial.. (shrink)

Scientific realism driven by inference to the best explanation (IBE) takes empirically confirmed objects to exist, independent, pace empiricism, of whether those objects are observable or not. This kind of realism, it has been claimed, does not need probabilistic reasoning to justify the claim that these objects exist. But I show that there are scientific contexts in which a non-probabilistic IBE-driven realism leads to a puzzle. Since IBE can be applied in scientific contexts in which empirical confirmation has (...) not yet been reached, realists will in these contexts be committed to the existence of empirically unconfirmed objects. As a consequence of such commitments, because they lack probabilistic features, the possible empirical confirmation of those objects is epistemically redundant with respect to realism. (shrink)

The aim of this paper is to argue for a revised and precisified version of the infamous Verifiability Criterion for the meaningfulness of declarative sentences. The argument is based on independently plausible premises concerning probabilistic confirmation and meaning as context-change potential, it is shown to be logically valid, and its ramifications for potential applications of the criterion are being discussed. Although the paper is not historical but systematic, the criterion thus vindicated will resemble the original one(s) in some important (...) ways. At the same time, it will also be more modest insofar as meaningfulness will turn out to be relativized linguistically and probabilistically, and different choices of the linguistic and probabilistic parameters may lead to different verdicts on meaningfulness. (shrink)

In this paper, I argue that causal theories of memory are typically committed to two independent, non-mutually entailing theses. The first thesis pertains to the necessity of appropriate causation in memory, specifying a condition token memories need to satisfy. The second pertains to the explanation of memory reliability in causal terms and it concerns memory as a type of mental state. Post-causal theories of memory can reject only the first (weak post-causalism) or both (strong post-causalism) theses. Upon this backdrop, I (...) examine Werning’s (2020) causalist argument from probabilistic correlation. I argue that it doesn’t establish the necessity of appropriate causation and thus it can only target strong post-causalist theories. I end up by presenting some general considerations, suggesting that memories may not always be causally linked to past experiences. (shrink)

Newcomb’s problem is a decision puzzle whose difficulty and interest stem from the fact that the possible outcomes are probabilistically dependent on, yet causally independent of, the agent’s options. The problem is named for its inventor, the physicist William Newcomb, but first appeared in print in a 1969 paper by Robert Nozick [12]. Closely related to, though less well-known than, the Prisoners’ Dilemma, it has been the subject of intense debate in the philosophical literature. After three decades, the issues remain (...) unresolved. Newcomb’s problem is of genuine importance because it poses a challenge to the theoretical adequacy of orthodox Bayesian decision theory. It has led both to the development of causal decision theory and to efforts aimed at defending the adequacy of the orthodox theory. (shrink)

The chapter explains why evolutionary genetics – a mathematical body of theory developed since the 1910s – eventually got to deal with culture: the frequency dynamics of genes like “the lactase gene” in populations cannot be correctly modeled without including social transmission. While the body of theory requires specific justifications, for example meticulous legitimations of describing culture in terms of traits, the body of theory is an immensely valuable scientific instrument, not only for its modeling power but also for the (...) amount of work that has been necessary to build, maintain, and expand it. A brief history of evolutionary genetics is told to demonstrate such patrimony, and to emphasize the importance and accumulation of statistical knowledge therein. The probabilistic nature of genotypes, phenogenotypes and population phenomena is also touched upon. Although evolutionary genetics is actually composed by distinct and partially independent traditions, the most important mathematical object of evolutionary genetics is the Mendelian space, and evolutionary genetics is mostly the daring study of trajectories of alleles in a population that explores that space. The ‘body’ is scientific wealth that can be invested in studying every situation that happens to turn out suitable to be modeled as a Mendelian population, or as a modified Mendelian population, or as a population of continuously varying individuals with an underlying Mendelian basis. Mathematical tinkering and justification are two halves of the mutual adjustment between the body of theory and the new domain of culture. Some works in current literature overstate justification, misrepresenting the relationship between body of theory and domain, and hindering interdisciplinary dialogue. (shrink)

I set up two axiomatic theories of inductive support within the framework of Kolmogorovian probability theory. I call these theories ‘Popperian theories of inductive support’ because I think that their specific axioms express the core meaning of the word ‘inductive support’ as used by Popper (and, presumably, by many others, including some inductivists). As is to be expected from Popperian theories of inductive support, the main theorem of each of them is an anti-induction theorem, the stronger one of them saying, (...) in fact, that the relation of inductive support is identical with the empty relation. It seems to me that an axiomatic treatment of the idea(s) of inductive support within orthodox probability theory could be worthwhile for at least three reasons. Firstly, an axiomatic treatment demands from the builder of a theory of inductive support to state clearly in the form of specific axioms what he means by ‘inductive support’. Perhaps the discussion of the new anti-induction proofs of Karl Popper and David Miller would have been more fruitful if they had given an explicit definition of what inductive support is or should be. Secondly, an axiomatic treatment of the idea(s) of inductive support within Kolmogorovian probability theory might be accommodating to those philosophers who do not completely trust Popperian probability theory for having theorems which orthodox Kolmogorovian probability theory lacks; a transparent derivation of anti-induction theorems within a Kolmogorovian frame might bring additional persuasive power to the original anti-induction proofs of Popper and Miller, developed within the framework of Popperian probability theory. Thirdly, one of the main advantages of the axiomatic method is that it facilitates criticism of its products: the axiomatic theories. On the one hand, it is much easier than usual to check whether those statements which have been distinguished as theorems really are theorems of the theory under examination. On the other hand, after we have convinced ourselves that these statements are indeed theorems, we can take a critical look at the axioms—especially if we have a negative attitude towards one of the theorems. Since anti-induction theorems are not popular at all, the adequacy of some of the axioms they are derived from will certainly be doubted. If doubt should lead to a search for alternative axioms, sheer negative attitudes might develop into constructive criticism and even lead to new discoveries. -/- I proceed as follows. In section 1, I start with a small but sufficiently strong axiomatic theory of deductive dependence, closely following Popper and Miller (1987). In section 2, I extend that starting theory to an elementary Kolmogorovian theory of unconditional probability, which I extend, in section 3, to an elementary Kolmogorovian theory of conditional probability, which in its turn gets extended, in section 4, to a standard theory of probabilistic dependence, which also gets extended, in section 5, to a standard theory of probabilistic support, the main theorem of which will be a theorem about the incompatibility of probabilistic support and deductive independence. In section 6, I extend the theory of probabilistic support to a weak Popperian theory of inductive support, which I extend, in section 7, to a strong Popperian theory of inductive support. In section 8, I reconsider Popper's anti-inductivist theses in the light of the anti-induction theorems. I conclude the paper with a short discussion of possible objections to our anti-induction theorems, paying special attention to the topic of deductive relevance, which has so far been neglected in the discussion of the anti-induction proofs of Popper and Miller. (shrink)

Starting from a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of coherence. We give a representation of the conjoined conditional and we show that this new object is a conditional random quantity, whose set of possible values normally contains the probabilities assessed for the two conditional events. We examine some cases of logical dependencies, where the conjunction is a conditional event; moreover, we give the (...) lower and upper bounds on the conjunction. We also examine an apparent paradox concerning stochastic independence which can actually be explained in terms of uncorrelation. We briefly introduce the notions of disjunction and iterated conditioning and we show that the usual probabilistic properties still hold. (shrink)

In this work, we derive the standard formalism of quantum theory by analyzing the behavior of single-variable systems under measurements. These systems, with minimal information capacity, exhibit indeterministic behavior in independent measurements while yielding probabilistically predictable outcomes in dependent measurements. Enforcing probability conservation in the probability transformations leads to the derivation of the Born rule, which subsequently gives rise to the Hilbert space structure and the Schrödinger equation. Additionally, we show that preparing physical systems in coherent states —crucial for observing (...) quantum phenomena— effectively reduces the number of independent variables to one. This first-principles derivation of quantum theory from probability conservation in single-variable systems offers new insights into the physical meaning of quantum theory and clarifies its domain of applicability. (shrink)

Most philosophers of religion subscribe to some variety of religious realism: they believe that religious statements aim at capturing a mind-independent reality and are true precisely if they successfully do so. Curiously, most religious realists also believe that at least some of our religious beliefs are rationally justified. In this paper, I argue that these positions are actually at odds with each other. Religious realists should rather be religious skeptics. I first argue that realism always implies the possibility of our (...) beliefs being false. Then, I develop a Copernican argument to show that we have no reason to believe that we are in an epistemically privileged position where we could be confident that our particular belief is justified while all competing beliefs are not. This seems impossible for two reasons: (a) From a probabilistic point of view (adopting a principle of mediocrity), it is highly unlikely that current humans have the maximum of epistemic capacities possible. Rather, our ability to understand reality is limited, so we should expect at least some parts of religious reality to remain unknown to us. (b) From a deep-time perspective, humanity has not pondered religious questions for long. We are probably still in the early stages of our cognitive evolution, and so we should treat our religious beliefs as provisional and open to revision. (shrink)

The traditional No-Miracles Argument (TNMA) asserts that the novel predictive success of science would be a miracle, and thus too implausible to believe, if successful theories were not at least approximately true. The TNMA has come under fire in multiple ways, challenging each of its premises and its general argumentative structure. While the TNMA relies on explaining novel predictive success via the truth of the theories, we put forth a deductive version of the No-Miracles argument (DNMA) that avoids inference to (...) the best explanation entirely. Instead, a relatively simple empirical framework and a probabilistic analysis can accomplish the ambitious goals of the TNMA while entirely sidestepping its problems. This close-but-distinct argument has many independent strengths and comparatively few weaknesses. Indeed, objections tailored specifically to the DNMA reveal surprising insights into how exactly NMAs are neither circular nor question-begging, as has been widely speculated. (shrink)

As I head home from work, I’m not sure whether my daughter’s new bike is green, and I’m also not sure whether I’m on drugs that distort my color perception. One thing that I am sure about is that my attitudes towards those possibilities are evidentially independent of one another, in the sense that changing my confidence in one shouldn’t affect my confidence in the other. When I get home and see the bike it looks green, so I increase my (...) confidence that it is green. But something else has changed: now an increase in my confidence that I’m on color-drugs would undermine my confidence that the bike is green. Jonathan Weisberg and Jim Pryor argue that the preceding story is problematic for standard Bayesian accounts of perceptual learning. Due to the ‘rigidity’ of Conditionalization, a negative probabilistic correlation between two propositions cannot be introduced by updating on one of them. Hence if my beliefs about my own color-sobriety start out independent of my beliefs about the color of the bike, then they must remain independent after I have my perceptual experience and update accordingly. Weisberg takes this to be a reason to reject Conditionalization. I argue that this conclusion is too pessimistic: Conditionalization is only part of the Bayesian story of perceptual learning, and the other part needn’t preserve independence. Hence Bayesian accounts of perceptual learning are perfectly consistent with potential underminers for perceptual beliefs. (shrink)

I begin by giving reasons to accept the block-universe view, the strongly supported by physics view that we live in a four-dimensional world. According to it, the past and the future are as real as the present. As a result, it seems that the future is determined in the sense that what will be the case will necessarily be the case. In the dissertation, I examine whether we have to accept this consequence. I show that we do not have to (...) bite the bullet: the future might be both real and contingent. I first start to consider modal realism (possible futures are real) as a possible solution. However, I propose then another account, actualist (positing the reality of only one possible world: the space-time we live in). This solution relies on a conventionalist theory about metaphysical modality (Sidelle, 1989). It states that modal modality is purely conventional. In combination with a realist interpretation of natural modality (causal and/or nomological), this framework allows me to propose that the future is conventionally closed and naturally contingent. In this view, the necessity of the future holds in virtue of linguistic conventions and then is conventionally necessary. But this is a distinct phenomenon from the natural contingency of the future: the future is naturally contingent because there are mind-independent probabilistic relations holding between the present and the future. This solution has strong consequences: most importantly, an anti-realist view about ordinary objects and physical particles. I end up by presenting and arguing in favour of the general image of the world that comes out from these views. (shrink)

Socialist buildings in the city of Kruja (Albania) date back after the Second World War between the years 1945-1990. These buildings were built during the time of the socialist Albanian dictatorship and the totalitarian communist regime. A questionnaire with 30 questions was conducted and 14 people were interviewed. The interviewed residents belong to a certain area of the city of Kruja. Based on the results obtained, diagrams have been conceived and mathematical regression models have been developed which will serve as (...) a base for drawing conclusions. The purpose of this study is to find a relationship that specifies the role of the inhabitants, the role of the dwellings, the physical-mechanical characteristics, and their energy efficiency, in order to improve the living conditions of the inhabitants. The independent variables interact with each other assuming positive or negative values depending on the probabilistic mathematical regression, giving different results based on the calculated data. The variables that co-exist among them are living space, quality of life, time spent at home, social interaction of residents, dwelling humidity level, dwelling orientation, satisfaction level of the inhabitants, dwelling improvements demand, and monthly electricity bills. (shrink)

Is the common cause principle merely one of a set of useful heuristics for discovering causal relations, or is it rather a piece of heavy duty metaphysics, capable of grounding the direction of causation itself? Since the principle was introduced in Reichenbach’s groundbreaking work The Direction of Time (1956), there have been a series of attempts to pursue the latter program—to take the probabilistic relationships constitutive of the principle of the common cause and use them to ground the direction (...) of causation. These attempts have not all explicitly appealed to the principle as originally formulated; it has also appeared in the guise of independence conditions, counterfactual overdetermination, and, in the causal modelling literature, as the causal markov condition. In this paper, I identify a set of difficulties for grounding the asymmetry of causation on the principle and its descendents. The first difficulty, concerning what I call the vertical placement of causation, consists of a tension between considerations that drive towards the macroscopic scale, and considerations that drive towards the microscopic scale—the worry is that these considerations cannot both be comfortably accommodated. The second difficulty consists of a novel potential counterexample to the principle based on the familiar Einstein Podolsky Rosen (EPR) correlations in quantum mechanics. (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...) relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

This paper concerns the semantic difference between strong and weak neces-sity modals. First we identify a number of explananda: their well-known in-tuitive difference in strength between ‘must’ and ‘ought’ as well as differ-ences in connections to probabilistic considerations and acts of requiring and recommending. Here we argue that important extant analyses of the se-mantic differences, though tailored to account for some of these aspects, fail to account for all. We proceed to suggest that the difference between ’ought’ and ’must’ (...) lies in how they relate to scalar and binary standards. Briefly put, must(φ) says that among the relevant alternatives, φ is selected by the relevant binary standard, whereas ought(φ) says that among the relevant al-ternatives, φ is selected by the relevant scale. Given independently plausi-ble assumptions about how standards are provided by context, this ex-plains the relevant differences discussed. (shrink)

In the aftermath of Gettier’s examples, knowledge came to be thought of as what you would have if in addition to a true belief and your favorite epistemic goody, such as justifiedness, you also were ungettiered, and the theory of knowledge was frequently equated, especially by its detractors, with the project of pinning down that extra bit. It would follow that knowledge contributes something distinctive that makes it indispensable in our pantheon of epistemic concepts only if avoiding gettierization has a (...) value that can be explained without presupposing the value of knowledge. Tracking-type knowledge has a value that no other logically possible conditions on true belief does. As an Evolutionarily Stable Strategy it preserves appropriate belief states through time and changing circumstances. If we characterize gettierization through the concept of relevance matching, then we see that avoiding gettierization has a value independent of that of knowledge, namely, understanding, and that it is unnecessary to add a clause to the tracking conditions to make them suppress gettierization directly, though fallibly. The bright line of value is between gettierization avoidance and understanding on the one hand and knowledge on the other, and so should be the bright line defining concepts. The concept of relevance matching is key to a definition of what it is to understand why p is true, as opposed merely to knowing that p is true. Perfect tracking implies perfect relevance matching, so knowledge and understanding are intimately connected but understanding also requires that one own states that accomplish the relevance matching rather than achieving it vicariously. The theory of understanding based on relevance matching implies that understanding requires appreciation of not only p but its connections to other matters, and explains how it is possible to know that p is true without understanding why. The view implies that understanding is literally simulation, and is suggestive about understanding other minds. (shrink)

Disagreement is a ubiquitous feature of human life, and philosophers have dutifully attended to it. One important question related to disagreement is epistemological: How does a rational person change her beliefs (if at all) in light of disagreement from others? The typical methodology for answering this question is to endorse a steadfast or conciliatory disagreement norm (and not both) on a priori grounds and selected intuitive cases. In this paper, I argue that this methodology is misguided. Instead, a thoroughgoingly Bayesian (...) strategy is what's needed. Such a strategy provides conciliatory norms in appropriate cases and steadfast norms in appropriate cases. I argue, further, that the few extant efforts to address disagreement in the Bayesian spirit are laudable but uncompelling. A modelling, rather than a functional, approach gets us the right norms and is highly general, allowing the epistemologist to deal with (1) multiple epistemic interlocutors, (2) epistemic superiors and inferiors (i.e. not just epistemic peers), and (3) dependence between interlocutors. (shrink)

The concept of causation plays a central role in many philosophical theories, and yet no account of causation has gained widespread acceptance among those who have investigated its foundations. Theories based on laws, counterfactuals, physical processes, and probabilistic dependence and independence relations (the list is by no means exhaustive) have all received detailed treatment in recent years—and, while no account has been entirely successful, it is generally agreed that the concept has been greatly clarified by the attempts. In (...) this magnificent book, Woodward aims to give a unified account of causation and causal explanation in terms of the notion of a manipulation (or intervention, terms which can be read interchangeably). Not only does he produce in my view the most illuminating and comprehensive account of causation on offer, his theory also opens a great many avenues for future work in the area, and has ramifications for many other areas of philosophy. Making Things Happen ought to be of interest not only to philosophers of causation and philosophers of science, but to any philosopher whose concerns involve assumptions about the nature of causation, laws, or explanation. (shrink)

How can different individuals' probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events -- the agenda -- is a sigma-algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain (...) or an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are sigma-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we compare our results with existing results on binary judgment aggregation and Arrovian preference aggregation. This paper is the first of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

The aim of the paper is to develop general criteria of argumentative validity and adequacy for probabilistic arguments on the basis of the epistemological approach to argumentation. In this approach, as in most other approaches to argumentation, proabilistic arguments have been neglected somewhat. Nonetheless, criteria for several special types of probabilistic arguments have been developed, in particular by Richard Feldman and Christoph Lumer. In the first part (sects. 2-5) the epistemological basis of probabilistic arguments is discussed. With (...) regard to the philosophical interpretation of probabilities a new subjectivist, epistemic interpretation is proposed, which identifies probabilities with tendencies of evidence (sect. 2). After drawing the conclusions of this interpretation with respect to the syntactic features of the probability concept, e.g. one variable referring to the data base (sect. 3), the justification of basic probabilities (priors) by judgements of relative frequency (sect. 4) and the justification of derivative probabilities by means of the probability calculus are explained (sect. 5). The core of the paper is the definition of '(argumentatively) valid derivative probabilistic arguments', which provides exact conditions for epistemically good probabilistic arguments, together with conditions for the adequate use of such arguments for the aim of rationally convincing an addressee (sect. 6). Finally, some measures for improving the applicability of probabilistic reasoning are proposed (sect. 7). (shrink)

In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can know (...) a mathematical proposition based solely on a probabilistic proof, one is then forced to admit that such an epistemic subject can know several lottery propositions based solely on probabilistic evidence. Insofar as knowledge of lottery propositions on the basis of probabilistic evidence alone is denied by the vast majority of epistemologists, it is concluded that this constitutes an epistemic reason for rejecting probabilistic proofs as a means of acquiring mathematical knowledge. (shrink)

In this paper I expound an argument which seems to establish that probabilism and special relativity are incompatible. I examine the argument critically, and consider its implications for interpretative problems of quantum theory, and for theoretical physics as a whole.

Are probabilism and special relativity compatible? Dieks argues that they are. But the possible universe he specifies, designed to exemplify both probabilism and special relativity, either incorporates a universal “now”, or amounts to a many world universe, or fails to have any one definite overall Minkowskian-type space-time structure. Probabilism and special relativity appear to be incompatible after all. What is at issue is not whether “the flow of time” can be reconciled with special relativity, but rather whether explicitly probabilistic (...) versions of quantum theory should be rejected because of incompatibility with special relativity. (shrink)

This paper examines a promising probabilistic theory of singular causation developed by David Lewis. I argue that Lewis' theory must be made more sophisticated to deal with certain counterexamples involving pre-emption. These counterexamples appear to show that in the usual case singular causation requires an unbroken causal process to link cause with effect. I propose a new probabilistic account of singular causation, within the framework developed by Lewis, which captures this intuition.

In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also (...) consider what this says about the similarities between mathematics and, on the one hand natural sciences, and on the other hand philosophy. (shrink)